Five Divisible Sets On Some Algebraic Surfaces

The branch curve of a generic triple covering is an irreducible curve of even degree with only cusp as its singularities.We can use the concept of 3-divisible set to study the cusps of a curve and a surface,and we can classify generic triple coverings by using 3-divisible sets.We will introduce the definition and properties of 3-divisible sets,and explain the relationship between 3-divisible sets and cyclic triple coverings.We define similarly the concept of 5-divisible sets of singular points of type A4 on a surface.By generalizing the relationship of 3-torsion group and ternary codes over F3,we establish the relationship between 5-torsion groups and codes over F5.We construct a code corresponding to a 5-divisible set to obtain some properties on 5-divisible sets,calculate the invariants of cyclic quintic covers,and bound the number of singular points in a 5-divisible set.We discuss in detail the properties of singular points of type A4 on K3 surfaces and normal quintic surfaces.In particular,we also classify the generic triple covering induced by a 3-divisible set on a plane curve of degree 18,and we prove that 57 cusps on this curve cannot form a 3-divisible set.